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This is because you are doing a rotation transformation, which is an SU(2) on the state vector (the vector the matrices act on), but when you do the transformation of the Pauli operators, it is a regular vector rotation of all three, so it preserves the length. This is a special case for 2 state quantum mechanics, any SU(2) transformation preserves the length of the coefficients of the Pauli matrix in the expansion of any 2 by 2 matrix, the Pauli matrices make an operator vector.